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We demonstrate that quasiperiodicity can radically change the ground stateproperties of 1D moir\'e systems with respect to their periodic counterparts.By studying an illustrative example we show that while narrow bands play asignificant role in enhancing interactions both for commensurate andincommensurate structures, only quasiperiodicity is able to extend the orderedphase down to an infinitesimal interaction strength. In this regime, thequasiperiodic-enabled state has contributions from infinitely many wavevectors. This quasi-factal regime cannot be stabilized in the commensurate caseeven in the presence of a narrow band. These findings suggest thatquasiperiodicity may be a critical factor in stabilizing non-trivial orderedphases in interacting moir\'e structures and signal out multifractalnon-interacting phases, recently found in 2D incommensurate moir\'e systems, asparticularly promising parent states.

Incommensurability enabled quasi-fractal order in 1D narrow-band moir\'e systems